2. Divide the number repeatedly by 2:
Keep track of each remainder.
We stop when we get a quotient that is equal to zero.
- division = quotient + remainder;
- 975 308 453 ÷ 2 = 487 654 226 + 1;
- 487 654 226 ÷ 2 = 243 827 113 + 0;
- 243 827 113 ÷ 2 = 121 913 556 + 1;
- 121 913 556 ÷ 2 = 60 956 778 + 0;
- 60 956 778 ÷ 2 = 30 478 389 + 0;
- 30 478 389 ÷ 2 = 15 239 194 + 1;
- 15 239 194 ÷ 2 = 7 619 597 + 0;
- 7 619 597 ÷ 2 = 3 809 798 + 1;
- 3 809 798 ÷ 2 = 1 904 899 + 0;
- 1 904 899 ÷ 2 = 952 449 + 1;
- 952 449 ÷ 2 = 476 224 + 1;
- 476 224 ÷ 2 = 238 112 + 0;
- 238 112 ÷ 2 = 119 056 + 0;
- 119 056 ÷ 2 = 59 528 + 0;
- 59 528 ÷ 2 = 29 764 + 0;
- 29 764 ÷ 2 = 14 882 + 0;
- 14 882 ÷ 2 = 7 441 + 0;
- 7 441 ÷ 2 = 3 720 + 1;
- 3 720 ÷ 2 = 1 860 + 0;
- 1 860 ÷ 2 = 930 + 0;
- 930 ÷ 2 = 465 + 0;
- 465 ÷ 2 = 232 + 1;
- 232 ÷ 2 = 116 + 0;
- 116 ÷ 2 = 58 + 0;
- 58 ÷ 2 = 29 + 0;
- 29 ÷ 2 = 14 + 1;
- 14 ÷ 2 = 7 + 0;
- 7 ÷ 2 = 3 + 1;
- 3 ÷ 2 = 1 + 1;
- 1 ÷ 2 = 0 + 1;
3. Construct the base 2 representation of the positive number:
Take all the remainders starting from the bottom of the list constructed above.
975 308 453(10) = 11 1010 0010 0010 0000 0110 1010 0101(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 30.
A signed binary's bit length must be equal to a power of 2, as of:
21 = 2; 22 = 4; 23 = 8; 24 = 16; 25 = 32; 26 = 64; ...
The first bit (the leftmost) indicates the sign:
0 = positive integer number, 1 = negative integer number
The least number that is:
1) a power of 2
2) and is larger than the actual length, 30,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 32.
5. Get the positive binary computer representation on 32 bits (4 Bytes):
If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 32.
975 308 453(10) = 0011 1010 0010 0010 0000 0110 1010 0101
6. Get the negative integer number representation. Part 1:
To write the negative integer number on 32 bits (4 Bytes),
as a signed binary in one's complement representation,
... replace all the bits on 0 with 1s and all the bits set on 1 with 0s.
Reverse the digits, flip the digits:
Replace the bits set on 0 with 1s and the bits set on 1 with 0s.
!(0011 1010 0010 0010 0000 0110 1010 0101)
= 1100 0101 1101 1101 1111 1001 0101 1010
7. Get the negative integer number representation. Part 2:
To write the negative integer number on 32 bits (4 Bytes),
as a signed binary in two's complement representation,
add 1 to the number calculated above
1100 0101 1101 1101 1111 1001 0101 1010
(to the signed binary in one's complement representation)
Binary addition carries on a value of 2:
0 + 0 = 0
0 + 1 = 1
1 + 1 = 10
1 + 10 = 11
1 + 11 = 100
Add 1 to the number calculated above
(to the signed binary number in one's complement representation):
-975 308 453 =
1100 0101 1101 1101 1111 1001 0101 1010 + 1
Number -975 308 453(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:
-975 308 453(10) = 1100 0101 1101 1101 1111 1001 0101 1011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.